Problem 82 - Algebra for College Students

Question

Express each of the following as a single fraction involving positive exponents only. \(x^{2} y^{-2}-x^{-1} y^{-3}\)

Step-by-Step Solution

Answer

The expression as a single fraction is \(\frac{x^3 - 1}{x y^3}\).

Step 1: Rewrite the expression with positive exponents

Start with the expression \(x^{2} y^{-2} - x^{-1} y^{-3}\). To express with positive exponents, apply the rule \(a^{-m} = \frac{1}{a^m}\). This means we rewrite each term as: \(x^{2} \cdot \frac{1}{y^{2}} - \frac{1}{x} \cdot \frac{1}{y^{3}}\), resulting in \(\frac{x^{2}}{y^{2}} - \frac{1}{x y^{3}}\).

Step 2: Identify a common denominator

To combine the fractions \(\frac{x^{2}}{y^{2}}\) and \(\frac{1}{x y^{3}}\), we need a common denominator. The denominators are \(y^2\) and \(x y^3\). The least common multiple is \(x y^3\).

Step 3: Rewrite each fraction with the common denominator

Transform the first fraction \(\frac{x^{2}}{y^{2}}\) to have the denominator \(x y^3\) by multiplying the numerator and denominator by \(x\), resulting in \(\frac{x^3}{x y^3}\). The second fraction \(\frac{1}{x y^3}\) already has the common denominator, so it remains as is: \(\frac{1}{x y^3}\).

Step 4: Combine the fractions

Now that both fractions have the same denominator \(x y^3\), combine them: \(\frac{x^3}{x y^3} - \frac{1}{x y^3} = \frac{x^3 - 1}{x y^3}\). This is the expression as a single fraction with positive exponents.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition or subtraction. In the expression given \(x^{2} y^{-2} - x^{-1} y^{-3}\), variables \(x\) and \(y\) are used alongside exponents. These exponents tell us how many times to multiply the base by itself.
An important part of working with algebraic expressions is dealing with exponents.

• Positive exponents indicate regular multiplication (e.g., \(x^2 = x \times x\)).
• Negative exponents represent reciprocal values (e.g., \(y^{-2} = \frac{1}{y^2}\)).

When handling negative exponents, it's crucial to convert them to positive by using the reciprocal. This conversion aligns with the goal of simplifying expressions, making them easier to manipulate in calculations. For instance, in our exercise, writing everything with positive exponents simplifies the task of finding a common denominator.

Finding the Common Denominator

To add or subtract fractions, they must share the same denominator. The denominator is the bottom part of a fraction, representing the total number of equal parts.
Finding a common denominator involves identifying a multiple that all denominators can divide into evenly. For the fractions \(\frac{x^{2}}{y^{2}}\) and \(\frac{1}{x y^{3}}\), the denominators are \(y^2\) and \(x y^3\). Their least common multiple (LCM) becomes the shared denominator.

• Determine the LCM of the denominators: Consider the highest powers of all variables present.
• The LCM for \(y^2\) and \(x y^3\) is \(x y^3\), as it includes both \(x\) and the higher power of \(y\).

Once the common denominator \(x y^3\) is identified, each fraction can be rewritten to reflect this shared base. This process is essential for accurately combining the terms.

Mastering Fraction Operations

Fraction operations involve adding, subtracting, multiplying, or dividing fractions. Each operation has its own set of rules, especially regarding the denominators.
When subtracting fractions like in our exercise, with expressions \(\frac{x^{3}}{x y^{3}}\) and \(\frac{1}{x y^{3}}\), ensure they have the same denominator.

• Add or subtract the numerators: With the common denominator \(x y^3\), subtract the numerators \(x^3\) and \(1\).
• Keep the denominator: Combine the numerators to get \(\frac{x^3 - 1}{x y^3}\).

This ensures the fractions are combined into a single, simplified expression. Understanding how to operate on fractions with positive exponents ensures effective manipulation of algebraic expressions, simplifying complex equations into manageable steps.